FIG. 9 illustrates elements of a typical communication system. The communication system includes a transmitter, a receiver, and a connecting medium called a channel. The transmitter prepares and sends data down the channel, and the receiver reflects the inverse operations of those in the transmitter in order to recover the transmitted data. In modem design, such as Pulse Code Modulation (i.e. “PCM”), the channel can include noise. For instance, the channel can include digital impairment and an additive independent noise “n”. Under all current modem design, the noise is considered independent of the signal transmitted by the transmitter.
The noise n is a continuous random variable with a probability density function fn(a). As a result, the received random variable Y is also of the continuous type. For a particular observed value, say Y(k), assume that the receiver determines the transmitted message is t(i). Then, the conditional probability of correct detection is just the probability that t(i) was actually transmitted given that Y(k) is observed. According to well known probability theory, the decision rule is to set the receiver output to t(i) if and only if:P[t(i)/Y=Y(k)]=maxn{P[t(n)/Y=Y(k)]} for all n. 
Thus, the optimum receiver is a probability processor. The optimum receiver computes the a posteriori probability given Y=Y(k) for all messages in the set and decides on the message with the largest computed a posteriori probability.
Further communication theory has demonstrated that functionality of the probability processor can be simplified for channels with arbitrary signal and noise statistics. In particular, receivers in the prior art assume that noise is independent of the signal characteristics. Under these circumstances, known receivers are designed according to a minimum distance criteria. In other words, the receiver decides on the message t(i) whose voltage level is closest to the received voltage level Y; this is referred to as minimum distance decoding.
FIG. 10 illustrates the minimum distance criteria as typically implemented. The transmitted signal levels are shown on the left, the received signal levels are shown on the right, and the digital impairment is shown between the transmitted and received signal levels. Under this deterministic communication system, each transmitted level is mapped onto a received signal level by the receiver. Received signal voltages that don't exactly line-up with a received signal level are mapped onto the closest signal level under minimum distance decoding.
FIG. 11 further illustrates an exemplary constellation design for a receiver using minimum distance decoding. When a communication system transmits “i” equally likely messages, the received voltage when the ith message is transmitted is Y=s(i)+n. Under minimum distance decoding the receiver chooses a message t(i) whose voltage level is closest to the received voltage level Y. Hence, if Y is less than Δ/2 then the receiver decides t1; if Δ/2<Y<3Δ/2 then the receiver decides t2; if 3Δ/2<Y<5Δ/2 then the receiver decides t3, etc. As illustrated by this example, the known constellation designs in a receiver include a constant range for each possible message level. With particular reference to minimum distance decoding, each possible message level has a constant range Δ.
FIG. 12 shows an example of the conditional probability density functions (i.e. “pdf”) for the different transmitted messages t1–t4. The probability density functions for each message do not end at the boundaries between messages t1–t4, rather the pdfs overlap. The overlapping of the pdfs gives rise to errors in the minimum distance decoding receiver of FIG. 11. In particular, a receiver can decide that message t3 was transmitted when message t4 was actually transmitted. These errors in the minimum distance decoding receiver can arise because current models assume that noise in the channel is independent of signal strength.
Accordingly, there is a need for a receiver that accounts for the interrelationship between noise in the channel and the transmitted signal.